The smashing spectrum of sheaves
Abstract
For an arbitrary $\infty$-topos, we classify the smashing localizations in the $\infty$-category of sheaves valued in derived vector spaces: Any of them is the restriction functor to a (unique) closed subtopos. Our proof is based on the existence of a Boolean cover. This result in particular gives us the first example of a nonzero presentably symmetric monoidal stable $\infty$-category whose smashing spectrum has no points. Combining this with the sheaves-spectrum adjunction, we obtain a Tannaka-type categorical reconstruction result for locales.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.03969
- arXiv:
- arXiv:2406.03969
- Bibcode:
- 2024arXiv240603969A
- Keywords:
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- Mathematics - Category Theory;
- Mathematics - Commutative Algebra;
- Mathematics - Algebraic Topology
- E-Print:
- 9 pages